Approximation of sign-regular kernels
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符号正则核的低秩逼近
- 研究问题
- 如何在降低参数化积分核的秩时控制一致逼近误差，并使误差控制同时适用于参数取值及总变差受限的积分测度？
- 方法与思路
- 作者借助逼近论中的 n-widths 工具，对矩形域上的严格符号正则核计算秩约束误差下界，并构造相应的低秩逼近。
- 材料陈述的贡献
- 摘要称所考察实例中的构造在舍入误差内达到下界，且一致误差可明显小于同秩 Taylor、奇异函数及 Chebfun2 逼近。
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原文摘要
Parameterized integrals, q y = ∫ K x , y d P x , are common in economic applications. To optimize a parameterized integral, or to relate one such integral to others, it is helpful to reduce the rank of the kernel K while controlling approximation error. A bound on the uniform approximation error of K x , y ≈ ∑ i = 1 n f i x g i y also bounds the uniform approximation error of q y ≈ ∑ i = 1 n c i g i y = ∑ i = 1 n ∫ f i x d P x g i y over all y and all signed measures P whose total variation does not exceed a fixed bound (such as probability measures), which may be useful in optimizing q or in relating q to other parameterized integrals. Bounding the mean squared error or the local error in approximating K generally does not bound the uniform approximation error of q . Many economically interesting kernels of parameterized integrals, including exp c x y , the Gaussian probability density function, and a wide class of Green’s functions, satisfy a condition known as strict sign-regularity . For K x , y strictly sign-regular on a rectangular domain x ∈ x L , x U , y ∈ y L , y U , I introduce a new method to efficiently compute a lower bound on the uniform error achievable by any rank- n approximation. I also provide a novel method to construct a rank- n approximation that numerically achieves the lower bound in every example I have examined, so in each such example my new method solves, to within rounding error, (1) inf f i , g i i = 1 n sup x ∈ x L , x U , y ∈ y L , y U K x , y − ∑ i = 1 n f i x g i y . My approach uses tools from the literature on n -widths in approximation theory (as summarized by Pinkus (1985)). I show that my new method’s uniform error can be orders of magnitude smaller than that of a Taylor series with the same rank. It also outperforms singular function approximations and the Chebfun2 approach of Townsend and Trefethen (2013) in uniform error, typically by wide margins. I describe several applications that demonstrate the practical utility of my approximation method.